Manifolds whose higher odd order curvature operators have constant eigenvalues at the basepoint

LetR ^k be the higher-order curvature operator of a Riemannian manifoldM. IfR ^k has constant eigenvalues at the basepointP ofM for all oddk, then all the odd covariant derivatives of the curvature tensor vanish atP. If the metric is real analytic, the geodesic involution is a local isometry atP.